Miscellaneous News

Harmonic Series Calculator

I wrote a program that computes countably many partials of any harmonic series.

Christoph Renhart's Harmonic Series Calculator
Getting an arbitrarily large amount of partials of a freely chosen fundamental is now just one click away.

How to get countably many partials of any tone you like in a very quick way

This program is dedicated to all friends of spectral music as well as composers and music theorists who have spent hours of their precious time writing down harmonic series of a given fundamental, calculate all the partials meticulously using a pocket calculator and comparing two or more overtone series on a piece of sheet music. Here’s the good news: There’s no need to make things more complicated than they should be. Let’s get all the numeracy done by using the computer. Remember the meaning of the word «computer»—computing things belongs to its core skills. And it keeps computing things flawlessly.

What can be done with this program? Suppose we would like to get the first 32 partials of the tone E2. Select E as tempered fundamental and 2 from the list of octaves (remember: C1 is the lowest C on the piano). Next click on Calculate and find the results below. In the table of results you will see the exact frequecy of each partial as well as its pitch (e.g. the 7th partial of a tone with a fundamental frequency of 32.703 Hz (C1) is 228.921 Hz which is A#3 minus 31 Cents).

Naturally all the frequencies depend significantly on the standard pitch. So we might want to compare the 17th partial on A2 using 440 Hz as a concert pitch with the 17th partial of A2 using 443 Hz as standard pitch. This can also done easily with the program. Let’s do the 440 Hz first:

  1. Enter 440 Hz as standard pitch
  2. Select A as tempered fundamental
  3. Select 2 as octave
  4. Click Calculate

Next, let’s run the program again with a different standard pitch:

  1. Enter 443 Hz as standard pitch
  2. Select A as tempered fundamental
  3. Select 2 as octave
  4. Click Calculate

Finally, compare the two results:

Fundamental A2 — 440 Hz standard pitch:

1110.000A2 plus 0 Cents
2220.000A3 plus 0 Cents
3330.000E4 plus 2 Cents
4440.000A4 plus 0 Cents
5550.000C#5 minus 14 Cents
6660.000E5 plus 2 Cents
7770.000G5 minus 31 Cents
8880.000A5 plus 0 Cents
9990.000B5 plus 4 Cents
101100.000C#6 minus 14 Cents
111210.000D#6 minus 49 Cents
121320.000E6 plus 2 Cents
131430.000F6 plus 41 Cents
141540.000G6 minus 31 Cents
151650.000G#6 minus 12 Cents
161760.000A6 plus 0 Cents
171870.000A#6 plus 5 Cents
Fundamental: A2 — results for 440 Hz as standard pitch

Fundamental A2 — 443 Hz standard pitch:

1110.750A2 plus 0 Cents
2221.500A3 plus 0 Cents
3332.250E4 plus 2 Cents
4443.000A4 plus 0 Cents
5553.750C#5 minus 14 Cents
6664.500E5 plus 2 Cents
7775.250G5 minus 31 Cents
8886.000A5 plus 0 Cents
9996.750B5 plus 4 Cents
101107.500C#6 minus 14 Cents
111218.250D#6 minus 49 Cents
121329.000E6 plus 2 Cents
131439.750F6 plus 41 Cents
141550.500G6 minus 31 Cents
151661.250G#6 minus 12 Cents
161772.000A6 plus 0 Cents
171882.750A#6 plus 5 Cents
Fundamental: A2 — results for 443 Hz as standard pitch

As you can see, the frequencies differ from each other. With 440 Hz as concert pitch the 17th partial of A2 is 1870 Hz (A#6 + 5 Ct.), whereas using a concert pitch of 443 Hz the 17th partial of A2 is 1882.75 Hz (also A#6 + 5 Ct. of course).

Moreover, you can also choose a non-tempered fundamental frequency in relation to any concert pitch you like. Let’s say, for instance, our concert pitch is 443 Hz and our fundamental is 60 Hz. As the fundamental is (possibly) not equal to a tempered tone, we select Other from the list of pitches and also Other from the list of octaves (it was somehow difficult to find an appropriate name for the input field and perhaps I’m gonna rename it as Other is admittingly not ideal here). Finally enter 60 in the field Fundamental / Grundfrequenz. Choose any amount of partials and click Calculate. Let’s have a look at the first 32 partials:

160.000A#1 plus 39 Cents
2120.000A#2 plus 39 Cents
3180.000F3 plus 41 Cents
4240.000A#3 plus 39 Cents
5300.000D4 plus 25 Cents
6360.000F4 plus 41 Cents
7420.000G#4 plus 8 Cents
8480.000A#4 plus 39 Cents
9540.000C5 plus 43 Cents
10600.000D5 plus 25 Cents
11660.000E5 minus 10 Cents
12720.000F5 plus 41 Cents
13780.000G5 minus 21 Cents
14840.000G#5 plus 8 Cents
15900.000A5 plus 27 Cents
16960.000A#5 plus 39 Cents
171020.000B5 plus 44 Cents
181080.000C6 plus 43 Cents
191140.000C#6 plus 36 Cents
201200.000D6 plus 25 Cents
211260.000D#6 plus 10 Cents
221320.000E6 minus 10 Cents
231380.000F6 minus 33 Cents
241440.000F6 plus 41 Cents
251500.000F#6 plus 12 Cents
261560.000G6 minus 21 Cents
271620.000G6 plus 45 Cents
281680.000G#6 plus 8 Cents
291740.000A6 minus 32 Cents
301800.000A6 plus 27 Cents
311860.000A#6 minus 16 Cents
321920.000A#6 plus 39 Cents
Fundamental: 60 Hz — results for 443 Hz as standard pitch

That’s it. Try it out and feel free to use it whenever you need to work with harmonic series.

Link to the calculator:

Special thanks to Daniel Mayer for checking the results with a SuperCollider patch and for some great advise to make the program’s interface more user-friendly!